1. Field of the Invention
The present invention generally relates to solid state laser oscillators and, more particularly, is directed to a solid state laser apparatus in which a fundamental wave laser beam produced in a laser medium is resonated so as to pass a nonlinear optical crystal element within a resonator to thereby produce a type II secondary harmonic laser beam.
2. Description of the Prior Art
A solid state laser oscillator is provided in the prior art to emit a laser beam of short wavelength by producing a secondary harmonic laser beam having frequency twice as high as that of a fundamental wave laser beam produced within a resonator of a solid state laser oscillator. Japanese Utility Model Laid-Open Publication No. 48-93784 describes such conventional solid state laser oscillator.
In this kind of solid state laser oscillator, the secondary harmonic laser beam is phase-matched with the fundamental wave laser beam within the nonlinear optical crystal element provided in the inside of the resonator including a laser medium, whereby the secondary harmonic laser beam can be produced efficiently.
In order to realize the phase-matching, the phase matching condition of the type I or type II must be established between the fundamental wave laser beam and the secondary harmonic laser beam.
The type I phase matching is based on the principle that one photon having twice as high a frequency is produced from two photons polarized in the same direction by utilizing ordinary ray of fundamental wave laser beam as shown by the following equation (1) expressed as: ##EQU1## Therefore, if the fundamental wave laser beam is polarized and introduced under the condition such that the polarization direction of the fundamental wave laser beam is matched with the direction of the nonlinear optical crystal element by utilizing a polarizer such as a polarizing beam splitter or the like, in principle then the phase of the polarized components (p-wave component and s-wave component, which are referred to as intrinsic polarizations) of the fundamental wave laser beam emitted from the nonlinear optical crystal element can be prevented from being changed. Thus, the generation of the secondary harmonic laser beam can be stably continued by the fundamental wave laser beam which oscillates within the resonator.
Whereas, in the type II phase matching, two orthogonally polarized intrinsic fundamental wave beams are introduced into the nonlinear optical crystal element, whereby the phase matching conditions are established between the two intrinsic polarizations. Therefore, the fundamental wave laser beam is divided into ordinary ray and extraordinary ray in the inside of the nonlinear optical crystal element, to thereby cause the phase matching in the extraordinary ray of the secondary harmonic laser beam as shown by the following equation (2): ##EQU2##
In the equations (1) and (2), n.sub.0(w) and n.sub..theta.(w) represent the refractive indexes of the fundamental wave laser beams (frequency f=w) relative to the ordinary ray and the extraordinary ray and n.sub.0(2w) and n.sub..theta.(2w) represent refractive indexes of the secondary harmonic laser beam (frequency f=2w) relative to the ordinary ray and the extraordinary ray.
A conventional solid state laser oscillator in which the nonlinear optical element for performing the type II phase matching is provided within the resonator will be described with reference to FIG. 1 (see pp. 1175 to 1176 of [Large-Amplitude fluctuations due to longitudinal mode coupling in diode-pumped intracavity-doubled Nd:YAG lasers]written by T. Baer and published by Journal of Optical Society of America Inc., Vol. 3, No. 9/Sep. 1986/J.Opt.Soc.Am.B).
Referring to FIG. 1, there is seen a laser diode 1 which emits a laser beam of wavelength of 808 nm and whose power is 200 mW. The diverged laser beam from this laser diode 1 is collimated by a collimator lens (convex lens) 14 and converged by an objective lens 15.
A YAG laser rod 4 is provided as the aforenoted laser medium and a dichroic mirror D, treated by the vapor deposition method, is attached to the laser rod 4 at its rear end face opposing the objective lens 15. Incidentally, the dichroic mirror D permits the incident light from the objective lens 15-side to be passed therethrough and reflects the incident light from the opposite side, that is, from the front end side direction of the laser rod 4. One end face of the laser rod 4 is formed as a curved face having a converging lens effect.
The converged light from the objective lens 15 is introduced into the laser rod 4 (i.e., optical pumping) and focused at a point p so that the laser rod 4 emits an infrared light having a wavelength of 1064 nm (this infrared light having the wavelength of 1064 nm will hereinafter referred to as a fundamental wave light). A nonlinear optical crystal element 6 is made of KTP (KTiOPO.sub.4), that is, uniaxial crystal having one optical axis and a cube whose one side is 5 mm.
The optical crystal element 6 has an incident loss of about 0.5% when the wavelength of incident light is 1064 nm. Also, this optical crystal element 6 permits the laser beams having wavelength of 532 nm and 1064 nm to pass therethrough and further matches the phases of the fundamental wave light (1064 nm) and the secondary harmonic laser beam (output light) having the wavelength of 532 nm (type II phase matching). A concave mirror 3 is a dichroic concave mirror having high reflectivity (99.9%) for a laser beam having a wavelength of 1064 nm and which has high transmissivity (98%) of a laser beam having a wavelength of 532 nm.
The fundamental wave light from the laser rod 4 is reciprocated between the dichroic mirror D formed at the rear end face of the laser rod 4 and the concave mirror 3 (the length therebetween is 60 mm). In accordance with the reciprocation of the fundamental wave light, due to the pull-in phenomenon, the phase of the reciprocated light is arranged and then amplified (i.e., induced emission) so that the laser beam is oscillated, that is, the oscillation having the wavelength of 1064 nm occurs. An output level of a green laser beam (wavelength is 532 nm) of SHG (secondary harmonic generation or 1/2 multiple wavelength light) having a wavelength of 1064 nm falls in a range of from about 5 to 10 mW.
The fundamental wave light derived from the above laser rod 4 by the optical pumping is vertically introduced into the plane of the optical crystal element 6. The incident fundamental wave light is divided into two straight polarized components (ordinary ray and extraordinary ray) which are vibrated in the directions perpendicular to each other within the planes perpendicular to the travelling direction. Accordingly, each time the fundamental wave light derived from the laser rod 4 is reciprocated in the space within a resonator RS and is passed through the optical crystal element 6, the phases of the orthogonally polarized intrinsic beams (polarized beam formed of extraordinary ray component and polarized beam formed of ordinary ray component) are displaced from each other to cause a coupling so that energy is exchanged between the two polarized beams. As a result, outputs of the extraordinary ray and of the ordinary ray are fluctuated from a time standpoint and a noise is produced. Accordingly, a stable and strong resonation state in which a laser beam of 532 nm wavelength is obtained cannot be formed and a conversion efficiency at which the resonance wave light is converted into a laser beam having a wavelength of 532 nm is low.
As described above, in the example of the prior art of FIG. 1, when the secondary harmonic laser beam is generated by using the type II phase matching condition, the phases of the intrinsic polarized beams of the fundamental wave laser beam are fluctuated each time the fundamental laser beam is repeatedly travelled through the nonlinear optical crystal element. There is then the risk that the generation of the secondary harmonic laser beam cannot be continued stably.
More specifically, if the phases of the orthogonally polarized intrinsic beams (i.e., p-wave component and s-wave component) are respectively displaced from each other each time the fundamental wave laser beam generated in the laser medium is repeatedly travelled through the nonlinear optical crystal element by the resonating operation, then a stationary state in which the fundamental wave laser beams efficiently intensify with each other in the respective portions of the resonator cannot be obtained, so that the strong resonance state (i.e., strong standing wave) cannot be formed. In consequence, the conversion efficiency in which the fundamental wave laser beam is converted into the secondary harmonic laser beam is deteriorated and there is then the risk that a noise will occur in the secondary harmonic laser beam.
Therefore, a solid state laser oscillator (laser light source) is proposed, in which the fundamental wave laser beam is stably resonated within the resonator under the condition that the type II phase matching condition is satisfied. This solid state laser oscillator is disclosed in Japanese Patent Laid-Open Publication No. 1-220879.
This conventional solid laser oscillator will be described with reference to FIG. 2. In FIG. 2, like parts corresponding to those of FIG. 1 are marked with the same references and therefore need not be described in detail.
As shown in FIG. 2, this solid state laser oscillator includes the Nd:YAG laser rod (laser medium) 4 which produces the fundamental wave laser light LA.sub.(w) by receiving the excitation laser beam emitted from the laser diode 1 to the light incident surface thereof through the collimator lens 14 and the objective lens 15.
This fundamental wave laser beam LA.sub.(w) is travelled through the nonlinear optical crystal element 6 formed of KTP (KTiOPO.sub.4), for example, a quarter-wave plate formed of a crystal plate and a birefringence element 16, in that order, and reflected on the reflecting plane of the concave mirror (dichroic mirror) 3. Then, the reflected fundamental wave laser beam is again travelled through the birefringence element 16, the nonlinear optical crystal element 6 and the laser rod 4 and then reflected on the reflecting plane (dichroic mirror) D of the laser rod 4.
Thus, the fundamental wave laser beam LA.sub.(w) is resonated so as to reciprocate in the resonance optical path formed between the reflecting plane (dichroic mirror) D of the laser rod medium 4 and the reflecting plane of the concave mirror 3, thereby constructing the resonator RS between the reflecting plane D and the concave mirror 3.
The birefringence element 16 is set at the optical axis position such that the direction of the extraordinary ray direction refractive index n.sub..theta.(7) is inclined by an azimuth angle of .theta.=45 degrees relative to the direction of the extraordinary ray direction refractive index n.sub..theta.(6) within the plane vertical to the propagation direction of light as shown in FIG. 3.
With the above-mentioned arrangement, when the fundamental wave laser light LA.sub.(w) travels through the nonlinear optical crystal element 6 via the resonant optical path, this fundamental wave laser light LA.sub.(w) produces the secondary harmonic laser beam LA.sub.(2w). This secondary harmonic laser beam LA.sub.(2w) is travelled through the concave mirror 3 and transmitted as an output laser beam LA.sub.OUT.
Under this condition, respective rays constituting the fundamental laser beam LA.sub.(w) are travelled through the birefringence element 16 located at the position displaced by the azimuth angle of .theta.=45 degrees relative to the nonlinear optical crystal element 6, thereby the power of the laser beams in the respective portions of the resonator are stabilized to a predetermined level.
Experimental results in the example of the prior art shown in FIG. 2 will be described as follows.
More specifically, into the resonator RS in which the Nd:YAG laser rod 4 is excited by the laser diode 1, the birefringence element 16 formed of the quarter-wave plate was inserted relative to the wavelength of the fundamental wave laser beam LA.sub.(w) (wavelength is 1.06 .mu.m) of the nonlinear optical crystal element 6 formed of KTP (KTiOPO.sub.4) and the resonator RS.
With this arrangement, under the conditions that the birefringence element 16 is placed at the position of the azimuth angle .theta.=0 degree (i.e., the optical axis of the extraordinary ray direction of the birefringence element 16 is made coincident with the optical axis of the extraordinary ray direction of the nonlinear optical crystal element 6) and that the azimuth angle .theta. is rotated by 45 degrees (i.e., .theta.=45 degrees), an extraordinary ray component E.sub..theta.(w) and an ordinary ray component E.sub.0(w) of the fundamental wave laser beam LA.sub.(w) and the secondary harmonic laser beam LA.sub.(2w) were respectively detected by a photodetector.
As a result, the extraordinary ray component E.sub..theta.(w) and the ordinary ray component E.sub.0(w) of the fundamental wave laser beam LA.sub.(w) in the first state where .theta.=0 degree demonstrated unstable changes with time t as shown in FIGS. 4A and 4B.
It was understood that the extraordinary ray component E.sub..theta.(w) and the ordinary ray component E.sub.0(w) have correlation therebetween because of the occurrence of mode competition.
Further, it was understood from FIG. 4C that the output P.sub.(2w) of the secondary harmonic laser beam LA.sub.(2w) produced in response to the fundamental laser beam LA.sub.(w) whose power level is unstably fluctuated with time t presents unstable fluctuation such as when the power level thereof is considerably fluctuated in a range from the high frequency component to the low frequency component.
Whereas, it was understood that, under the second state that the azimuth angle .theta. of the birefringence 16 is set to .theta.=45 degrees, the extraordinary ray component E.sub..theta.(w) and the ordinary ray component E.sub.0(w) of the fundamental wave laser beam LA.sub.(w) are stabilized so as to present substantially constant values with time t as shown in FIG. 5A and 5B. Also, it was understood that the output P.sub.(2w) of the secondary harmonic laser beam LA.sub.(2w) produced by the stabilized fundamental wave laser beam LA.sub.(w) is stabilized to substantially a constant value as shown in FIG. 5C.
Since the fundamental wave laser beam LA.sub.(w) resonating through the resonant optical path is not rectilinearly polarized by the polarizing element or the like, the fundamental wave laser beam LA.sub.(w) brings the two orthogonally polarized two intrinsic beams in the fundamental wave mode, and these two intrinsic polarized beams become random polarized beams which are free from the phase relation between the two modes.
When the secondary harmonic laser beam LA.sub.(2w) is produced within the nonlinear optical crystal element 6 by such fundamental wave laser beam LA.sub.(w), then the output P.sub.(2w) becomes proportional to a product of the extraordinary ray component output P.sub..theta.(w) and the ordinary ray component output P.sub.0(w) of the fundamental wave laser beam LA.sub.(w) within the nonlinear optical crystal element 6 as expressed by the following equation (3) EQU P.sub.(2w) .varies.d.sup.2 .multidot.P.sub..theta.(w) .multidot.PO.sub.(w) ( 3)
where d.sup.2 is the proportional constant.
However, when the output P.sub.(2w) is expressed by the product of the extraordinary ray component output P.sub..theta.(w) and the ordinary ray component output P.sub.0(w), a coupling occurs between the two intrinsic polarizations (i.e., polarizations formed of the extraordinary ray component and of the ordinary ray component) so that the energy is interchanged between the two polarizations.
If the energy is interchanged between the two polarizations, i.e., the extraordinary ray component and the ordinary ray component within the nonlinear optical crystal element 6, the outputs P.sub..theta.(w) and P.sub.0(w) of the extraordinary ray component and the ordinary ray component are fluctuated with time t, thus resulting in the secondary harmonic output P.sub.(2w) generated in the non-linear optical crystal element 6 being made unstable.
That is, in the arrangement in which the azimuth angle .theta. of the birefringence element 16 is selected to be .theta.=0 degree, the output laser beam LA.sub.OUT contains a noise component of very large energy which cannot be used in actual practice as shown in FIG. 6A.
It is to be noted that the noise spectrum of the output laser beam LA.sub.OUT contains a noise of about 53 dB when the frequency f is about 5 MHz, as shown by a curve K1 in FIG. 6B.
While the azimuth angle .theta. of the birefringence element 16 is set to .theta.=45 degrees, it was understood that the output laser beam LA.sub.OUT provides a stabilized signal whose noise component is sufficiently suppressed as shown in FIG. 7A. Also, it was understood that the noise spectrum thereof was improved in S/N (signal-to-noise) ratio by about 80 dB when the frequency f is 5 MHz as shown by a curve K2 in FIG. 7B.
It will be clear from the above experimental results that, according to the solid state laser apparatus of FIG. 2, when the secondary harmonic laser beam LA.sub.(2w) is generated within the nonlinear optical crystal element 6 under the type II phase matching condition, the azimuth angle .theta. of the birefringence element 16 is selected to be 45 degrees so that the coupling phenomenon can be prevented from being produced between two propagations of the fundamental laser beam LA.sub.(w) which propagates the resonance optical path of the resonator RS. In consequence, the output laser beam LA.sub.OUT formed of the secondary harmonic laser beam LA.sub.(2w) can be stabilized.
Therefore, since the orthogonally polarized two intrinsic beams within the resonance optical path of the resonator are brought in the fundamental wave mode and the fundamental wave laser beam LA.sub.(w) of the randomly polarized beam having no correlation with the phase relation between the two modes can be resonated, the extra polarizer need not be interposed, thus simplifying the whole arrangement more.
When the secondary harmonic laser beam LA.sub.(2w) is generated within the nonlinear optical crystal element 6 under the type II phase matching condition as shown in FIG. 2, the resonance operation is stabilized by inserting the birefringence element 16 into the azumuth angle position of the azimuth angle .theta. of 45 degrees. The reason for this is understood from a theoretical standpoint as follows:
That is, within the resonator RS, the following rate equations are established when the above two modes are established: ##EQU3## EQU .tau..sub.f =-(.beta.I.sub.2 .beta..sub.21 I.sub.1 +1)G.sub.2 +G.sub.2.sup.0 ( 7)
where .tau..sub.o is the reciprocating time of the resonator, .tau..sub.f is the fluorescent life, .alpha..sub.1 and .alpha..sub.2 the loss coefficients in the two modes, respectively, .epsilon..sub.1 is the loss coefficient caused by the occurrence of the secondary harmonic component in each mode, .epsilon..sub.2 is the loss coefficient caused by the occurrence of sum frequency between the two modes, .beta. is the saturation parameter, G.sub.1.sup.0 and G.sub.2.sup.0 are the small signal gains in the two modes, respectively, I.sub.1 and I.sub.2 the light intensities in the two modes, respectively, G.sub.1 and G.sub.2 are the gains in the two modes, respectively, and .beta..sub.12 and .beta..sub.21 the cross-saturation parameters in the two modes, respectively.
In association with the above rate equations, there is known a paper which pointed out that the resonating operation of the resonator becomes unstable because of the coupling in the multilongitudinal modes. That is, the rate equation with respect to the coupling between the multilongitudinal modes is described in PP. 1175 to 1180 of "Large-Amplitude fluctuations due to longitudinal mode coupling in diode-pumped intracavity-doubled Nd:YAG lasers" written by T. Baer and published by Journal of Optical Society of America Inc. Vol. 3, No. 9/Sep. 1986/J.Opt. Soc. Am.B.
The rate equation in this paper can be similarly applied to the two intrinsic polarization modes so that the equations (4) to (7) can be established with respect to the two intrinsic polarization modes.
Of the equations (4) to (7), the equations (4) and (6) involve a multiplying term (-2.sub..epsilon.2 I.sub.1 I.sub.2) which have the light intensities I.sub.1 and I.sub.2 of the two intrinsic polarization modes, whereby the light intensities of the two intrinsic polarization modes in the inside of the resonator are coupled with each other. In this connection, the equations (4) and (6) express the relation such that when the light intensity I.sub.1 (or I.sub.2) is fluctuated, the light intensity I.sub.2 (or I.sub.1) is also fluctuated.
However, the coefficient .epsilon..sub.2 in the multiplying term of -2.sub..epsilon.2 I.sub.1 I.sub.2 satisfies .epsilon..sub.2 =0 when the azimuth angle .theta. is selected so as to satisfy .theta.=45 degrees. Whereas, when the azimuth angle .theta. is selected to be .theta..noteq.45 degrees, the coefficient .epsilon..sub.2 takes the values other than 0, which can be proved as follows. Under this condition, the multiplying term of -2.sub..epsilon.2 I.sub.1 I.sub.2 can be erased from the rate equations (4) and (6) so that the resonating operations expressed by the equations (4) and (6) can be stabilized.
Initially, let us consider that .theta.=0 degree is selected as one example of the general conditions where the azimuth angle .theta. is selected so as to satisfy .theta..noteq.45 degrees.
Electric field vectors E.sub.1 and E.sub.2 of two intrinsic polarization of beams incident on the nonlinear optical crystal element 6 at that time are incident on the nonlinear optical crystal element 6 under the condition such that they become coincident with the ordinary ray o and the extraordinary ray e of the nonlinear optical crystal element 6. Accordingly, expressing the incident electric field vectors E.sub.1 and E.sub.2 by Jones vector where the ordinary ray axis o of the nonlinear optical crystal element 6 is taken as an x axis and the extraordinary ray axis e is taken as y axis yields the following equations (8) and (9): ##EQU4## where Jones vector is expressed only by coefficients omitting the phase term.
Thus, time average value P.sub.(w) of the fundamental wave laser beam LA.sub.(w) within the resonator CAV can be expressed as the sum of the square of the magnitudes E.sub.1 and E.sub.2 of electric fields by the following equation (10): ##EQU5## where (E.sub.1 +E.sub.2)* and E.sub.1 + and E.sub.2 * represent conjugated vectors of (E.sub.1 +E.sub.2), E.sub.1 and E.sub.2, respectively.
In the equation (10), when terms to be multiplied with each other have values having strong correlation, or in the case of E.sub.1 and E.sub.2, time average values E.sub.1 * and E.sub.2 * are expressed by the following equations (11) and (12): EQU E.sub.1 E.sub.1 *=.vertline.E.sub.1 .vertline..sup.2 .tbd.P.sub.1 ( 11) EQU E.sub.2 E.sub.2 *=.vertline.E.sub.2 .vertline..sup.2 .tbd.P.sub.2 ( 12)
Whereas, in the case of E.sub.1 E.sub.2 * and E.sub.2 E.sub.1 *, the electric fields E.sub.1 and E.sub.2 expressed by the multiplying terms are, respectively, electric field components of the two intrinsic orthogonally polarized modes. Further, due to random polarization having no correlation for the phase relation between the two modes, they lost correlation therebetween and in consequence, the time values become 0 as expressed by the following equations (13) and (14): EQU E.sub.1 E.sub.2 *=0 (13) EQU E.sub.2 E.sub.1 *=0 (14)
The electric field E.sub.(2w) of the secondary harmonic laser beam LA.sub.(2w) can be expressed, in the case of the type II phase matching, by the following equation (15): EQU E.sub.(2w) =dE.sub.1 E.sub.2 ( 15)
where d is the nonlinear conversion efficiency of the nonlinear optical crystal element 6.
The time average value P.sub.(2w) of the power of the secondary harmonic laser beam LA.sub.(2w) can be expressed by the product of powers of the two intrinsic polarizations: ##EQU6## Also in this case, the relationships expressed by the equations (11) to (14) can be established.
Therefore, when the azimuth angle .theta. is selected to be zero degree, the power of the resonator becomes the sum of the power P.sub.1 +P.sub.2 of the fundamental wave laser beam LA.sub.(w) expressed with respect to the equation (10) and the power d.sub.2 P.sub.1 P.sub.2 of the secondary harmonic laser beam LA.sub.(2w) expressed by the equation (16).
Comparing this relationship with the equations (4) and (6), it is to be noted that the light intensities I.sub.1 and I.sub.2 in the equations (4) and (6) have the same meaning as those of the powers P.sub.1 and P.sub.2 in the equations (10) and (16) and that the equation (4) includes the term [i.e., (G.sub.1 -.alpha..sub.1)I.sub.1 ] of the light intensity I.sub.1, the term (i.e., -.sub..epsilon. I.sub.1 2) of I.sub.1.sup.2 and the multiplying term (i.e., -2.sub..epsilon.2 I.sub.1 I.sub.2) of I.sub.1 and I.sub.2 and the equation (6) includes the term (i.e., .sub..epsilon.1 I.sub.2.sup.2) of I.sub.2 and the multiplying term (i.e., -2.sub..epsilon.2 I.sub.1 I.sub.2) of I.sub.1 I.sub.2.
Accordingly, it is to be understood that, when .epsilon..sub.1 is selected to be zero (.epsilon..sub.1 =0) in the equations (4) and (6), the sum of the equations (4) and (6) has the same term as the sum of the equations (10) and (16).
Thus, when the azimuth angle .theta. of the birefringence element 7 is selected to be zero (.theta.=0 degree), this setting is equivalent to the fact that the constant .epsilon..sub.1 is set to zero (.epsilon..sub.1 =0.degree.) in the general equations of the equations (4) and (6). However, when the azimuth angle .theta. is selected to be 0 degree (.theta.=0), the multiplying term -2.sub..epsilon.2 I.sub.1 I.sub.2 of the light intensities I.sub.1 and I.sub.2 of the two fundamental wave modes cannot be erased because .epsilon..sub.2 .noteq.0. Accordingly, when the azimuth angle .theta. is selected to be zero (.theta.=0.degree.) as shown in FIG. 8, the resonant operation of the resonator expressed by the rate equations of the equations (4) and (6) cannot be stabilized.
In the solid state laser resonator shown in FIG. 2, when the azimuth angle .theta. of the birefringence element 16 is selected to be 45 degrees (.theta.=45.degree.), this means that, as shown in FIG. 9, the intrinsic polarizations E.sub.1 and E.sub.2 of the fundamental wave laser beam LA.sub.(w) within the resonator are set to the azimuth angle positions which are rotated relative to the ordinary ray axis o and the extraordinary ray axis e of the nonlinear optical crystal element 6. This can be proved by the equation (17) because of the following reasons.
As a result, the intrinsic vectors E1 and E2 can be expressed by the following Jones vectors: ##EQU7##
Accordingly, the time average value P.sub.(w) of the power of the fundamental wave laser beam LA(w) of the resonator CAV can be expressed similarly to the equations (10) to (14) as follows: ##EQU8##
Whereas, the electric field E.sub.(2w) of the secondary harmonic laser beam LA.sub.(2w) generated under the type II phase matching condition is expressed by the following equation (20) with reference to the components of the ordinary ray axis o and the extraordinary ray axis e: ##EQU9##
From the equation (20), the time average value P.sub.(2w) of the power P.sub.(2w) of the secondary harmonic laser beam LA.sub.(2w) can be expressed as follows: ##EQU10## where the following equations are established: EQU E.sub.1.sup.2 E.sub.1.sup.*2 =.vertline.E.sub.1 .vertline..sup.4 .tbd.P.sub.1.sup.2 ( 22) EQU E.sub.2.sup.2 E.sub.2.sup.*2 =.vertline.E.sub.2 .vertline..sup.4 .tbd.P.sub.2.sup.2 ( 23) EQU E.sub.1.sup.2 E.sub.2.sup.*2 =0 (24) EQU E.sub.2.sup.2 E.sub.1.sup.*2 =0 (25)
In this connection, since the terms of E.sub.1.sup.2 E.sub.1.sup.*2 and E.sub.2 E.sub.2.sup.*2 of the equation (21) have the equations in which E.sub.1, E.sub.1.sup.* and E.sub.2, E.sub.2.sup.* each having strong correlation are multiplied, the resultant time average value does not become zero but becomes equal to the square of the powers P.sub.1 and P.sub.2.
Whereas, since the electric fields E.sub.1, E.sub.2.sup.* and E.sub.2, E.sub.1.sup.*2 are respective electric field components of the two orthogonally intrinsic polarization modes and are also the random polarizations which are not correlated with the phase relationship between the two modes, these electric fields are not correlated with each other, thereby the time average values of the terms of E.sub.1.sup.2 E.sub.2.sup.*2 and E.sub.2.sup.2 E.sub.1.sup.*2 being made zero.
Having compared the sum of the time average value P.sub.(w) (the equation (19)) of the power P.sub.(w) of the fundamental wave laser beam LA.sub.(w) produced when the azimuth angle .theta. of the birefringence element 16 is selected to be 45 degrees (.theta.=45.degree.) and the time average value P.sub.(2w) (the equation (21)) of the power P.sub.(2w) of the secondary harmonic laser beam LA.sub.(2w) with the sum of the equations (4) and (6), it is to be noted that, when the coefficient .epsilon..sub.2 of the multiplying terms of the light intensities I.sub.1 and I.sub.2 is selected to be zero (.epsilon..sub.2 =)) in the equations (4) and (6), the respective terms of the sum of the equations (19) and (20) correspond with the respective terms of the sum of the equations (4) and (6) in a one-to-one relation.
This means that the fact that the azimuth angle .theta. of the birefringence element 16 shown in FIG. 2 is selected to be 45 degrees (.theta.=45.degree.) is equivalent to the fact such that the coefficient .epsilon..sub.2 is selected to be zero (.epsilon..sub.2 =0) in the equations (4) and (6) expressed as the general equations. If the above-mentioned conditions are set, then it will be possible to obtain the resonant condition expressed by the rate equation which avoids the term expressed by the product of the respective light intensities I.sub.1 and I.sub.2 of the two fundamental wave modes in the equations (4) and (6), thereby inhibiting the energy from being interchanged through the occurrence of the secondary harmonic laser beam between the light intensities I.sub.1 and I.sub.2 of the two fundamental wave modes. Therefore, the secondary harmonic laser beam LA.sub.(2w) can be stabilized in accordance with the fundamental wave laser beam LA.sub.(w).
The above conditions can be established if the birefringence element 16 in which the azimuth angle .theta. is selected to be 45 degrees (.theta.=45.degree.) and the phase amount .DELTA. is selected to be 90 degrees (.DELTA.=90.degree.) is selected.
That is, as shown in FIG. 10, if the phase is deviated by a phase amount .delta. by the birefringence when the fundamental wave laser beam LA.sub.(w) passes the nonlinear optical crystal element 6, then the corresponding polarization state will be expressed by Jones matrix C(.delta.) as in the following equation (26): ##EQU11##
Further, since the birefringence element 16 is rotated by the azimuth angle .theta., the polarization state of the fundamental wave laser beam LA.sub.(w) can be expressed by Jones matrix R(.theta.) as in the following equation (27) ##EQU12##
Further, the polarization state in which the fundamental laser beam LA.sub.(w) is optically rotated by the phase amount .DELTA. by the birefringence element 16 can be expressed by Jones matrix C(.DELTA.) as in the following equation (28): ##EQU13##
Therefore, the change of the polarization state in which the fundamental wave laser beam LA.sub.(w) emitted from the laser rod 4 travels sequentially through the nonlinear optical crystal element 6 and the birefringence element 16 and introduced into the incident surface of the concave mirror 3 and reflected by the incident surface to return through the birefringence element 16 and the nonlinear optical crystal element 6 to the laser rod 4 side can be expressed by Jones matrix M as in the following equation (29): EQU M=C(.delta.)R(.theta.)C(.DELTA.)C(.DELTA.)R(-.theta.)C(.delta.) (29)
Substituting the equations (26) to (28) into the equation (29) yields the following Jones matrix M which expresses the polarization state of the optical system: ##EQU14##
Performing operation under the condition such that Jones matrix of the second to fifth terms on the right side of the equation (29) is taken as matrix M1, we have: ##EQU15## Substituting the calculated result into the equation (29), we have: ##EQU16## Then, the matrix M expressing the polarized state is put as: ##EQU17## and an intrinsic value .lambda. relative to the intrinsic vector X is calculated: EQU MX=.lambda.X (34)
The intrinsic value .lambda. which satisfies the equation (34) should satisfy the following determinant (35): ##EQU18## Then, opening this determinant (35), we have: EQU (A-.lambda.)(D-.lambda.)-BC=0 (36) EQU .lambda..sup.2 -(A+D).lambda.+AD-BC=0 (37)
Accordingly, it is to be understood that the quadratic equation with respect to .lambda. may be solved. A solution of the equation (37) becomes: ##EQU19## From the equations (32) and (33), A+D can be arranged as: ##EQU20## Further, AD-BC can be arranged as in the following equation (40): ##EQU21## Accordingly, substituting the equations (39) and (40) into the equation (38), we have the intrinsic value .lambda. which is expressed as: ##EQU22##
Therefore, if the x component of the intrinsic vector X is set to 1 (x=1), the intrinsic vector X can be expressed as: ##EQU23## Incidentally, in the solid state laser apparatus shown in FIG. 2, the aximuth angle .theta. of the birefringence element 16 is selected to be 45 degrees (.theta.=45.degree.) and the phase angle .DELTA. of the birefringence element 16 is selected to be 90 degrees (.DELTA.=90.degree.). Accordingly, putting EQU .theta.=45.degree. (43) EQU .DELTA.=90.degree. (44)
into the equations (32) and (33) yields the following matrix M: ##EQU24## Also, the equation (41) yields the intrinsic value .lambda. expressed as: EQU .lambda.=.+-.i (46)
Consequently, the intrinsic vector x can be obtained as: ##EQU25##
It is clear from the above-mentioned results that, if the azimuth angle .theta. is selected to be 45 degrees (.theta.=45.degree.) and the phase amount .DELTA. is selected to be 90 degrees (.DELTA.=90.degree.) with respect to the equations (43) and (44) as described above, then this means that, when the intrinsic polarized vectors E.sub.1 and E.sub.2 of the fundamental wave laser beam LA.sub.(w) within the resonator CAV becomes incident on the nonlinear optical crystal element 6 from the laser medium 2 side, the intrinsic polarized vectors E.sub.1 and E.sub.2 are set at the azimuth angle positions which are rotated by 45 degrees relative to the ordinary ray axis o and the extraordinary ray axis e of the nonlinear optical crystal element 6.
As a result of the consideration from a theoretical standpoint, it is to be noted that the azimuth angle .theta. of the birefringence element 16 is selected to be 45 degrees (.theta.=45.degree.), whereby the secondary harmonic laser beam LA.sub.(2w) can be stabilized in accordance with the fundamental wave laser beam LA.sub.(w) of the resonator CAV.
According to the conventional solid state laser apparatus shown in FIG. 2, the fundamental laser beam generated in the laser medium (laser rod) 4 is resonated so as to pass the nonlinear optical crystal element 6 provided within the resonator RS, thereby generating the type II secondary harmonic laser beam. Also, since the optical means for suppressing the coupling produced due to the occurrence of the sum frequency between the two polarization modes of the fundamental laser beam, that is, the birefringence element (quarter-wave plate) 16 is provided within the resonator RS, the coupling produced due to the occurrence of the sum frequency between the two polarization modes of the fundamental laser beam can be suppressed, thus resulting in the oscillation being stabilized.
According to the prior-art solid state laser apparatus as described above, however, if the multilongitudinal mode exists within the same polarization mode of the fundamental laser beam, there is then the risk that a mode hopping noise will occur due to the mode coupling within the same polarization mode.